Subgroups — #15761536

Order 3328.

306 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13113588	C₂	1	2	yes	maximal
2	C₂×#11718050	C₂	1	1	yes	maximal, characteristic
3	#13113119	C₂	1	1	yes	maximal, characteristic
4	#13112687	C₂	1	2	yes	maximal
5	C₄×C₁₃×#108	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#108	C₄	1	1	yes	characteristic
7	C₂×C₁₃×#108	C₄	1	1	yes	characteristic
8	C₄×C₁₃×M₁₆	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	#11718586	C₂²	1	1	yes	characteristic
11	#11718586	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊#1055	C₂²	1	2	yes	—
13	C₁₃×#341	C₄	1	2	yes	—
14	#11718050	—	2	2	no	—
15	#11718050	—	2	1	no	—
16	#11718050	—	2	1	no	—
17	#11718049	—	2	2	no	—
18	#11718047	—	2	1	no	—
19	C₈×#1055	—	2	1	no	—
20	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
21	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
25	C₂²×#1055	—	2	2	no	—
26	C₁₃×#108	—	2	2	no	—
27	C₁₃×#108	C₈	1	4	yes	complemented
28	C₁₃×#97	—	2	2	no	—
29	C₄×C₈×C₁₃	—	2	2	no	—
30	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
31	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
33	C₄×C₈×C₁₃	Q₈	1	1	yes	characteristic
34	C₂²⋊#1055	—	2	1	no	—
35	C₂²⋊#1055	Q₈	1	2	yes	—
36	C₄⋊#1055	—	2	1	no	—
37	C₄⋊#1055	—	2	2	no	—
38	C₄⋊#1055	—	2	2	no	—
39	C₄⋊#1055	D₄	1	1	yes	characteristic
40	C₄⋊#1055	D₄	1	1	yes	characteristic
41	C₄×#1055	—	2	1	no	—
42	C₄×#1055	—	2	1	no	—
43	#9608	—	13	1	no	maximal, Sylow
44	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
47	C₁₃×M₁₆	—	2	2	no	—
48	C₁₃×M₁₆	—	2	2	no	—
49	C₁₃×M₁₆	—	2	2	no	—
50	C₁₃×M₁₆	C₂×C₈	1	4	yes	—
51	C₂×C₈×C₁₃	—	2	2	no	—
52	C₂×C₈×C₁₃	—	2	1	no	—
53	C₂×C₈×C₁₃	—	2	1	no	—
54	C₂×C₈×C₁₃	—	2	2	no	—
55	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
60	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
61	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
64	C₄²×C₁₃	#41	1	1	yes	characteristic
65	C₂×#1055	—	2	1	no	—
66	C₂×#1055	—	2	1	no	—
67	C₂×#1055	—	2	2	no	—
68	C₂×#1055	—	4	2	no	—
69	C₂×#1055	—	8	2	no	—
70	C₄×#108	D₁₃	1	1	yes	characteristic
71	#1565	—	13	2	no	—
72	C₂×#335	—	13	1	no	—
73	#1450	—	13	1	no	—
74	#1283	—	13	2	no	—
75	C₂³×C₁₃	C₄×Q₈	1	1	yes	characteristic
76	C₂×C₄×C₁₃	—	2	1	no	—
77	C₂×C₄×C₁₃	—	2	1	no	—
78	C₂×C₄×C₁₃	—	4	1	no	—
79	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
83	C₂×C₄×C₁₃	#125	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
87	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
88	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
89	C₈×C₁₃	—	2	1	no	—
90	C₈×C₁₃	—	2	1	no	—
91	C₈×C₁₃	—	2	4	no	—
92	C₈×C₁₃	—	4	2	no	—
93	C₈×C₁₃	C₄⋊C₈	1	2	yes	—
94	C₈×C₁₃	C₄⋊C₈	1	2	yes	—
95	#1055	—	4	2	no	—
96	#1055	—	4	1	no	—
97	#1055	—	4	1	no	—
98	#1055	—	8	2	no	—
99	#431	—	13	1	no	—
100	#431	—	13	1	no	—
101	C₂×#108	Dic₁₃	1	1	yes	characteristic
102	C₂×#108	Dic₁₃	1	1	yes	characteristic
103	C₂×C₄⋊C₈	—	13	2	no	—
104	C₄×M₁₆	C₂×D₁₃	1	2	yes	—
105	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
106	#341	Dic₁₃	1	2	yes	—
107	#335	—	26	2	no	—
108	#335	—	26	1	no	—
109	#335	—	26	1	no	—
110	#334	—	26	2	no	—
111	#322	—	26	1	no	—
112	C₈²	—	26	1	no	—
113	C₂²×C₁₃	—	2	1	no	—
114	C₂²×C₁₃	—	2	1	no	—
115	C₂²×C₁₃	#389	1	1	yes	characteristic
116	C₂²×C₁₃	C₈×Q₈	1	1	yes	characteristic
117	C₂²×C₁₃	#446	1	1	yes	characteristic
118	C₄×C₁₃	—	2	1	no	—
119	C₄×C₁₃	—	2	1	no	—
120	C₄×C₁₃	—	2	1	no	—
121	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
122	C₄×C₁₃	#424	1	1	yes	characteristic
123	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
124	C₄×C₁₃	#436	1	1	yes	characteristic
125	C₄×C₁₃	#472	1	1	yes	characteristic
126	C₄×C₁₃	#473	1	1	yes	characteristic
127	C₂×M₁₆	C₂×Dic₁₃	1	2	yes	—
128	C₂×M₁₆	C₂×Dic₁₃	1	2	yes	—
129	C₂²×C₈	—	26	2	no	—
130	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
131	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
132	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
133	#108	—	2	2	no	—
134	#108	#1055	1	4	yes	complemented
135	C₄⋊C₈	—	13	1	no	—
136	C₄⋊C₈	—	13	1	no	—
137	C₄⋊C₈	—	26	2	no	—
138	C₄⋊C₈	—	26	2	no	—
139	C₄⋊C₈	—	26	1	no	—
140	C₂²⋊C₈	—	13	2	no	—
141	C₂²⋊C₈	—	26	1	no	—
142	#97	—	2	2	no	—
143	C₄×C₈	—	2	2	no	—
144	C₄×C₈	—	26	1	no	—
145	C₄×C₈	—	26	1	no	—
146	C₄×C₈	Dic₂₆	1	1	yes	characteristic
147	C₄×C₈	Dic₂₆	1	1	yes	characteristic
148	C₄×C₈	D₅₂	1	1	yes	characteristic
149	C₄×C₈	D₅₂	1	1	yes	characteristic
150	C₂×C₁₃	—	2	1	no	—
151	C₂×C₁₃	—	2	1	no	—
152	C₂×C₁₃	#1939	1	1	yes	characteristic
153	C₂×C₁₃	#1952	1	1	yes	characteristic
154	C₂×C₁₃	#1953	1	1	yes	characteristic
155	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
156	C₂²×C₄	#6131	1	1	yes	characteristic
157	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
158	M₁₆	—	2	2	no	—
159	M₁₆	—	2	2	no	—
160	M₁₆	—	2	2	no	—
161	M₁₆	C₂×#1055	1	4	yes	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	26	1	no	—
167	C₂×C₈	—	26	1	no	—
168	C₂×C₈	—	26	2	no	—
169	C₂×C₈	—	52	2	no	—
170	C₂×C₈	—	104	2	no	—
171	C₂×C₈	C₂×#1055	1	2	yes	—
172	C₂×C₈	#6099	1	2	yes	—
173	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
174	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
175	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
176	C₂×C₈	C₄⋊Dic₁₃	1	1	yes	characteristic
177	C₄²	C₂×Dic₂₆	1	1	yes	characteristic
178	C₄²	C₂×D₅₂	1	1	yes	characteristic
179	C₄²	D₄×D₁₃	1	1	yes	characteristic
180	C₄²	#6129	1	1	yes	characteristic
181	C₁₃	#9608	1	1	yes	characteristic, complemented, Sylow
182	C₂³	Q₈×Dic₁₃	1	1	yes	characteristic
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	4	1	no	—
186	C₂×C₄	#88142	1	1	yes	characteristic
187	C₂×C₄	#88144	1	1	yes	characteristic
188	C₂×C₄	#88149	1	1	yes	characteristic
189	C₂×C₄	#88151	1	1	yes	characteristic
190	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
191	C₂×C₄	C₂×#6099	1	1	yes	characteristic
192	C₂×C₄	C₂×C₄⋊Dic₁₃	1	1	yes	characteristic
193	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
194	C₂×C₄	#88202	1	1	yes	characteristic
195	C₂×C₄	#88202	1	1	yes	characteristic
196	C₈	—	2	1	no	—
197	C₈	—	2	1	no	—
198	C₈	—	2	4	no	—
199	C₈	—	4	2	no	—
200	C₈	—	52	2	no	—
201	C₈	—	52	1	no	—
202	C₈	—	52	1	no	—
203	C₈	—	104	2	no	—
204	C₈	C₄⋊#1055	1	2	yes	—
205	C₈	C₄⋊#1055	1	2	yes	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	#11718646	1	1	yes	characteristic
209	C₂²	Q₈×#1055	1	1	yes	characteristic
210	C₂²	#11718685	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	#11718557	1	1	yes	characteristic
215	C₄	#11718558	1	1	yes	characteristic
216	C₄	C₂×C₄⋊#1055	1	1	yes	characteristic
217	C₄	#11718660	1	1	yes	characteristic
218	C₄	D₄×#1055	1	1	yes	characteristic
219	C₄	#11718673	1	1	yes	characteristic
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	#13115555	1	1	yes	characteristic
223	C₂	#13115622	1	1	yes	characteristic
224	C₂	#13115627	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88149
commutator	C₂×C₄×C₁₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₄×C₁₃×#108	C₂
socle	C₂²×C₁₃	#389
prime radical	C₂³×C₁₃	C₄×Q₈
cosocle	C₂×C₄²×C₁₃	C₂³
hypercenter	C₄×#108	D₁₃
hypocenter	C₁₃	#9608
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
27. C₁₃×#108	C₈	semidirect
134. #108	#1055	semidirect
181. C₁₃	#9608	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#13113588	2
C₂×#11718050	2
#13113119	2
#13112687	2
C₄×C₁₃×#108	2
#9608	13